Dynamic BigBinaryString implmentation in C++ - c++

Question

I am writing a BigBinaryString class in C++ which abstractly holds the large binary string and can perform operations like xor, and, left shift and right shift.
I have stored the binary string internally as a vector of unsigned longs, so each element in vector consumes 64-bits from the bitstring.
I have decided to use have four types of constructors:
BigBinaryString(const string bitstring) //for directly converting the bitstring to internal repr.
BigBinaryString() //initialize the binary string to 0.
BigBinaryString(const size_t num) //hold the binary for corresponding unsigned long.
BigBinaryString(const vector<size_t> vec) //Directly pass the internal repr to the constructor for it to copy.
So far I have implemented and, xor, equality and left shift operators.
However, I feel the left shift operator has a very high time complexity in which the way I have implemented it.
So, I need a few suggestions on how to speed up the left shift operator so that I can implement the right shift efficiently as well.
The code so far is as follows:
#define SIZE_UNSIGNED_LONG (sizeof(size_t) * 8)
class BigBinaryString {
private:
vector<size_t> num;
void reduce() {
while (*num.rbegin() == 0) {
num.pop_back();
}
if (num.size() == 0) {
num.push_back(0);
}
}
public:
BigBinaryString() { num.push_back(0); }
BigBinaryString(const size_t n) { num.push_back(n); }
BigBinaryString(const vector<size_t> vec) {
const size_t length = vec.size();
for (size_t i = 0; i < length; i++) {
num.push_back(vec.at(i));
}
reduce();
}
BigBinaryString operator&(const BigBinaryString& op) {
vector<size_t> vec;
size_t maxlen = max(num.size(), op.num.size());
size_t minlen = min(num.size(), op.num.size());
const size_t zero = 0;
for (size_t i = 0; i < minlen; i++) {
vec.push_back(num.at(i) & op.num.at(i));
}
return BigBinaryString(vec);
}
BigBinaryString(const string bitstring) {
string temp = bitstring;
size_t dec = 0;
while (temp.length() != 0) {
if (temp.length() > SIZE_UNSIGNED_LONG) {
dec = stoul(temp.substr(temp.length() - SIZE_UNSIGNED_LONG,
SIZE_UNSIGNED_LONG),
nullptr, 2);
temp = temp.substr(0, temp.length() - SIZE_UNSIGNED_LONG);
} else {
dec = stoul(temp, nullptr, 2);
temp = "";
}
num.push_back(dec);
}
reduce();
}
BigBinaryString operator^(const BigBinaryString& op) {
vector<size_t> vec;
size_t maxlen = max(num.size(), op.num.size());
size_t minlen = min(num.size(), op.num.size());
for (size_t i = 0; i < maxlen; i++) {
if (i < minlen) {
vec.push_back(num.at(i) ^ op.num.at(i));
} else if (maxlen == num.size()) {
vec.push_back(num.at(i));
} else if (maxlen == op.num.size()) {
vec.push_back(op.num.at(i));
}
}
return BigBinaryString(vec);
}
bool operator==(const BigBinaryString& op) {
if (num.size() != op.num.size()) {
return false;
}
size_t length = num.size();
for (size_t i = 0; i < length; i++) {
if (num.at(i) != op.num.at(i)) {
return false;
}
}
return true;
}
bool operator==(const size_t n) {
BigBinaryString op(n);
if (num.size() != op.num.size()) {
return false;
}
size_t length = num.size();
for (size_t i = 0; i < length; i++) {
if (num.at(i) != op.num.at(i)) {
return false;
}
}
return true;
}
bool operator!=(const BigBinaryString& op) { return not(*this == op); }
bool operator!=(const size_t op) { return not(*this == op); }
BigBinaryString operator<<(size_t shift) {
string bitstring = this->to_string();
bitstring.append(shift, '0');
return BigBinaryString(bitstring);
}
string to_string() {
string suffix = "";
string retval = "";
string prefix = "";
size_t n = 0;
for (auto i = num.rbegin(); i != num.rend(); i++) {
n = *i;
prefix.clear();
suffix.clear();
while (n != 0) {
suffix = (n % 2 == 0 ? "0" : "1") + suffix;
n /= 2;
}
if (i != num.rbegin()) {
prefix.append(SIZE_UNSIGNED_LONG - suffix.size(), '0');
}
prefix = prefix + suffix;
if (prefix.size() == SIZE_UNSIGNED_LONG) {
retval += prefix;
} else if (i == num.rbegin()) {
retval += prefix;
} else if (i != num.rbegin()) {
throw invalid_argument("prefix+suffix error");
}
}
return retval;
}
};
Any help will be appreciated!